UNIVERSITY OF CALIFORNIA, SAN DIEGO Linear Network Coding over Ring Alphabets A ...

200 Pages·2017·1.33 MB·English

UNIVERSITY OFCALIFORNIA, SAN DIEGO
LinearNetwork Coding overRing Alphabets
A dissertationsubmittedinpartial satisfactionofthe
requirements forthedegreeDoctorofPhilosophy
in
Electrical Engineering
(CommunicationTheory &Systems)
by
JosephMichaelConnelly
Committeeincharge:
ProfessorKennethZeger,Chair
ProfessorYoung-HanKim
ProfessorDanielRogalski
ProfessorPaulSiegel
ProfessorLanceSmall
2018
Copyright
JosephMichael Connelly,2018
Allrightsreserved.
The dissertation of Joseph Michael Connelly is approved, and it
isacceptable inqualityandformforpublication onmicrofilmand
electronically:
Chair
UniversityofCalifornia,San Diego
2018
iii
TABLEOFCONTENTS
SignaturePage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
TableofContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Vita. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
AbstractoftheDissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Chapter1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 NetworkCoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Group,Ring,andFieldAlphabets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 ScalarLinearCodesoverFiniteFields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 VectorLinearCodesoverFiniteFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 LinearNetworkCodingoverFiniteRings . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter2 ScalarCodesandCommutativeRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 NetworkModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 OurContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 RingDominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Fundamental RingComparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 MinimizingAlphabetSize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 DirectProductsofRings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.4 Then-Choose-TwoNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.5 RingsofSize p2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 FiniteFieldDominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 LocalRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Non-Power-of-PrimeSizeRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 IntegerPartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 PartitionDivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 Characterizing MaximalPartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.3 MaximalPartitionsofShortLength . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 MaximalCommutativeRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 MultipleMaximalRingsofaGivenSize. . . . . . . . . . . . . . . . . . . . . . . 47
2.6 OpenQuestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
iv
Chapter3 VectorCodesandNon-CommutativeRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1 LinearCodesOverModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.2 OurContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.3 Comparisons ofModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 CommutativeandNon-CommutativeRings . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.1 ModulesandVectorLinearCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 TheDim-kNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1 Insufficiency ofCommutativeRings. . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 ModulesoftheSameSize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4.1 CommutativeRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.2 Non-CommutativeRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.1 SummaryofResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5.2 OpenQuestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.A ProofsofLemmas3.4.12,3.4.13,and3.4.14 . . . . . . . . . . . . . . . . . . . . . . . . . 88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter4 CapacityandAchievableRateRegions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.1 Modules, LinearFunctions, andTensorProducts . . . . . . . . . . . . . . . . . 95
4.1.2 NetworkCodingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.3 LinearityoverFiniteRingsandModules. . . . . . . . . . . . . . . . . . . . . . . 101
4.1.4 RateRegions,Capacity,andSolvability . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.5 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1.6 MainResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2 Fractional andVectorCodesoverModules . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.1 Fractional EquivalentNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 Fractional Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2.3 MatrixRingsoverFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3 LinearRateRegionsoverFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3.1 ComparingLinearRateRegionsoverDifferentFields. . . . . . . . . . . . . . 115
4.4 LinearRateRegionsoverRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4.1 ComparingLinearCapacities overDifferentRings . . . . . . . . . . . . . . . . 119
4.4.2 AsymptoticSolvability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.5 Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Chapter5 AClassofNon-LinearlySolvableNetworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1.1 NetworkCodingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1.2 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
v
5.1.3 OurContributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.1.4 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 NetworkN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
0
5.3 NetworkN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
1
5.4 NetworkN (m,w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2
5.5 NetworkN (m ,m ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3 1 2
5.6 NetworkN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4
5.6.1 Solvability ofN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4
5.6.2 LinearSolvability ofN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4
5.6.3 CapacityandLinearCapacityofN (m) . . . . . . . . . . . . . . . . . . . . . . . 163
4
5.6.4 SizeofN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4
5.7 OpenQuestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.A CapacityProofsofN ,N andN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
1 2 3
5.A.1 N CapacityProof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
1
5.A.2 N CapacityProof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
2
5.A.3 N CapacityProof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Chapter6 BigPictureDiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.1 CanaNetworkbeLinearlySolvableoverRingsbutnotFields?. . . . . . . . . . . . . 187
6.2 Whatisthe“Best”AlphabetofaGivenSizeforLinearCoding? . . . . . . . . . . . . 188
6.3 Whatisthe“Best”AlphabetforLinearCodingonaGivenNetwork?. . . . . . . . . 188
6.4 OverWhatAlphabetSizesisaGivenNetworkSolvable? . . . . . . . . . . . . . . . . . 188
6.5 CantheLinearCapacityofaNetworkbeIncreased UsingRings? . . . . . . . . . . . 189
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LISTOFFIGURES
Figure1.1: TheButterflyNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Figure1.2: Then-Choose-TwoNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Figure1.3: TheM Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Figure1.4: TheDiabolicalNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Figure2.1: Asummaryoftheresultsinthischapterforafixednetwork . . . . . . . . . . . . . . . . . . 19
Figure2.2: Asummaryoftheresultsinthischapterforafixedalphabet size . . . . . . . . . . . . . . . 19
Figure2.3: TheChar-mNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Figure2.4: Then-Choose-TwoNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Figure2.5: TheTwo-SixNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure2.6: Themaximalpartitions ofk=1,2,...,30underpartition division. . . . . . . . . . . . . . . 50
Figure3.1: TheFanoNetwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Figure3.2: Then-Choose-TwoNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Figure3.3: TheDim-kNetwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figure3.4: TheM Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Figure3.5: Atrivialnetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Figure4.1: TheButterflyNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure4.2: The(k ,k ,n)-ButterflyNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
x y
Figure4.3: TheChar-mNetwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Figure5.1: Summaryofthenetworksconstructed inthischapter . . . . . . . . . . . . . . . . . . . . . . . 132
Figure5.2: Anetworkbuilding block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Figure5.3: ThenetworkN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
0
Figure5.4: ThenetworkN (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
1
Figure5.5: ThenetworkN (m,w). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
2
Figure5.6: ThenetworkN (m ,m ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3 1 2
vii
ACKNOWLEDGEMENTS
First and foremost, I would like to thank my advisor, Ken Zeger, for his guidance, support, and
inspiration. I am indebted to him for the many opportunities he has presented me with and for his efforts
to help me become a better researcher. Ken’s passion for research, teaching, and mathematics is highly
contagious, andIhopetomaintain asimilarenthusiasm throughout mycareer. Iamgrateful toYoung-Han
Kim,PaulSiegel,DanRogalski,andLanceSmallforserving onmycommitteeandtakingthetimetoread
andeditthisdissertation. IthankallofthefriendsIhavemadeandthosewhomIhaveworkedwithatUCSD
for making graduate school a memorable and rewarding experience. I am thankful to Richard Sahara and
MarcRiedelforbothinspiringandencouragingmetopursuegraduateeducation. IalsothankJonHamkins
forhismentorshipandhelpinmakingmytimeatJPLasenjoyable asitwas.
Myparentshaveprovidedmewiththemeansandtheencouragementtopursuemygoals,nomatter
wheretheytakeme. Iamthankful fortheirunconditional support throughout mylife, particularly overthe
past 5 years. Finally, I thank my fiance´e Allison Flickinger, for her patience and her willingness to put up
with me through the highs and lows of writing this dissertation. This work would not have been possible
wereitnotforherloveandunwaiveringsupport.
Thechaptersofthisdissertationconsistofpublishedandsubmittedjournalarticles. Thedissertation
authorwastheprimaryinvestigator andauthorofeachofthesepapers.
• Chapter2isareprintofthematerialasitappearsinJ.ConnellyandK.Zeger,“Linearnetworkcoding
over rings – Part I: Scalar codes and commutative alphabets,” IEEE Transactions on Information
Theory,vol. 64,no. 1,pp. 274–291,January2018.
• Chapter3isareprintofthematerialasitappearsinJ.ConnellyandK.Zeger,“Linearnetworkcoding
overrings–PartII:Vectorcodesandnon-commutativealphabets,”IEEETransactionsonInformation
Theory,vol. 64,no. 1,pp. 292–308,January2018.
• Chapter 4 has been submitted for publication of the material J. Connelly and K. Zeger, “Linear ca-
pacityofnetworksoverringalphabets.”
• Chapter5isareprintofthematerialasitappearsinJ.ConnellyandK.Zeger,“Aclassofnon-linearly
solvablenetworks,”IEEETransactionsonInformationTheory,vol. 63,no. 1,pp. 201–229,January
2017.
Thisworkwassupported, inpart,bytheNationalScienceFoundation.
viii
VITA
2013 BachelorofElectricalEngineering, UniversityofMinnesotaTwinCities
2013 BachelorofComputerEngineering, UniversityofMinnesotaTwinCities
2013–2018 TeachingAssistant,UniversityofCalifornia, SanDiego
2014–2017 GraduateStudentResearcher, UniversityofCalifornia, SanDiego
2016 Master of Science in Electrical Engineering (Communication Theory & Systems),
UniversityofCalifornia, SanDiego
2016 GraduateStudentIntern, NASAJetPropulsionLaboratory
2017 AssociateInstructor, UniversityofCalifornia, SanDiego
2018 DoctorofPhilosophyinElectricalEngineering(CommunicationTheory&Systems),
UniversityofCalifornia, SanDiego
2018– Developmental Engineer, AirForceResearchLabs,NewMexico
PUBLICATIONS
J. Connelly and K. Zeger, “A class of non-linearly solvable networks,” Proceedings of the IEEE Interna-
tionalSymposium onInformation Theory(ISIT),pp. 1964–1968, Barcelona, Spain,July10-152016.
J.Connelly, “Repeat-PPM super-symbol synchronization,” IPNProgress Report vol. 42, no. 207, Novem-
ber2016.
J. Connelly and K. Zeger, “A class of non-linearly solvable networks,” IEEE Transactions on Information
Theory,vol. 63,no. 1,pp. 201–229,January 2017.
J.ConnellyandK.Zeger,“Linearnetworkcodingoverrings–PartI:Scalarcodesandcommutativealpha-
bets,”IEEETransactions onInformation Theory,vol. 64,no. 1,pp. 274–291,January2018.
J. Connelly and K.Zeger, “Linear network coding overrings – PartII: Vector codes and non-commutative
alphabets,” IEEETransactions onInformation Theory,vol. 64,no. 1,pp. 292–308,January2018.
J. Connelly and K. Zeger, “Linear capacity of networks over ring alphabets,” submitted to IEEE Transac-
tionsonInformation Theory,June4,2017,revisedJanuary29,2018.
ix
ABSTRACT OF THEDISSERTATION
LinearNetwork Coding overRing Alphabets
by
JosephMichaelConnelly
DoctorofPhilosophyin ElectricalEngineering
(CommunicationTheory &Systems)
UniversityofCalifornia, San Diego,2018
ProfessorKennethZeger, Chair
As connected devices play an ever-growing role in our society, there is a subsequent need for ad-
vancesinmulti-usercommunicationsystems. Inanetwork,sendersandreceiversareconnectedviaaseries
ofintermediate users whoshare information represented assequences ofbitsorelements ofsomeother fi-
nitealphabet. Byallowinguserstotransmitfunctionsoftheirinputs,asopposedtosimplyrelayingreceived
data, the information throughput of a network can be increased. Network codes in which these functions
are linear are suboptimal in general but are of practical interest due to their mathematical tractability and
low implementation complexity. The study of linear network coding has primarily been limited to finite
field alphabets. In this work, we consider linear network codes over more general algebraically-structured
alphabets, namely finite rings. Wecontrast linear network codes over finite fields, commutative rings, and
non-commutative rings, and we discuss cases where non-linear codes attain higher information rates than
evenverygenerallinearcodes. Ourresultsshowthatfinitefieldsare,insomesense,thebestringalphabets
for linear network coding, but in certain instances, it may be advantageous to use linear coding over some
otherringalphabetofthesamesize. Specifically,weproveresultsrelatedto:
x